1

a

f[x_] := 4 x^4 + 5 x^5 - 6 x^6 + x^0 - 2/x^2

D[f[x], x]

D[f[x], x] /. x -> 1

ArcTan[D[f[x], x]] /. x -> 1

ArcTan[9] // N

ArcTan[9]/Degree // N

D[f[x], {x, 2}]

b

Integrate[f[x], x]

Integrate[f[x], {x, 1, 2}]

% // N

c

f[x_] := 1/(Sin[x/3 - 3])^2

D[f[x], x]

Integrate[f[x], x]

d

f[x_] := 1/(x (x - 2) (x + 2))

Integrate[f[x], x]

Integrate[f[x], {x, 3, 5}]

% // N

Integrate[f[x], {x, 3, Infinity}]

% // N

d

f[x_] := 7 Log[x] - (E^x - E^(-x))/E^(3  x)

D[f[x], x]

Integrate[f[x], x]

2

p1 = {0, 0, 0}; p2 = {1, 0, 0}; p4 = {0, 1, 0}; 
    p3 = {1, 1, 0};
p5 = {0, 0, 1}; p6 = {1, 0, 1}; p8 = {0, 1, 1}; 
    p7 = {1, 1, 1};

p12  = (p1 + p2 )/2; p23 = (p2 + p3 )/2; 
    p34 = (p3 + p4)/2; p41 = (p4 + p1)/2;
p56  = (p5 + p6 )/2; p67 = (p6 + p7 )/2; 
    p78 = (p7 + p8)/2; p85 = (p8 + p5)/2;
p15  = (p5 + p1 )/2; p26 = (p6 + p2 )/2; 
    p37 = (p7 + p3)/2; p48 = (p8 + p4)/2;
pm = (p56 + p67 + p26)/
          3;

p = 
        Graphics3D[{PointSize[0.02], Point[p1], Point[p1], Point[p2], 
            Point[p3], Point[p4], Point[p5], Point[p6], Point[p7], 
            Point[p8], 
Point[p12], Point[p23], Point[p34], 
            Point[p41], 
Point[p56], Point[p67], 
            Point[p78], Point[p85], 
Point[p15], 
            Point[p26], Point[p37], 
            Point[p48], 
Point[{0.5, 0.5, 
                0.5}], 
Point[pm]}, 
          Boxed -> False];
lin = 
        Graphics3D[{Line[{p4, p1, p2, p3, p4, p8, p7, p3, p7, p6, p2, p6, p5, 
                p1, p5, p8, p4}], Line[{p12, p23, p34, p41, p12}], 
            Line[{p12, p26, p56, p15, p12}], Line[{p23, p37, p67, p26, p23}], 
            Line[{p34, p37, p78, p48, p34}], Line[{p41, p15, p85, p48, p41}], 
            Line[{p56, p67, p78, p85, p56}], 
Line[{p1, 
                p7}], Line[{p4, p6}], Line[{p3, p5}], Line[{p2, p8}]}, 
          Boxed -> False];
Show[p, lin, 
        ViewPoint -> {2.686,  1.686,  0.877}];

a

vLen[a_] := Sqrt[a . a];
h = vLen[p6 - pm]

N[%]

V = Det[{p56 - p6, p67 - p6, p26 - p6}]/6

N[%]

b

8  V

N[%]

1 - 8  V

N[%]

c

ob = 3 V/h

N[%]

h1 = vLen[p56 - p67]

q = h1^2

obTotal = 8  ob + 6  q

N[%]

3

Plot[{18/5 t, 24/5 (t - 5/6) , 12, 15, 16}, {t, 0, 5}, 
        Epilog -> {Line[{{10/3, 0}, {10/3, 12}}], 
            Line[{{10/3 + 5/6, 0}, {10/3 + 5/6, 17}}]}];

s1[u_] := 
        Evaluate[Integrate[15/5 t, {t, 0, u}]];
s2[u_] := 
        Evaluate[Integrate[15/4.5 (t - 0.5), {t, 0.5, u}]];

Plot[{s2[u],  s1[u]}, {u, 0, 10}];

Solve[{vM t0 == 12, vH (t0 - 5/6) == 12,  
        vM (t0 + 5/6) == vH (t0 + 5/6 - 5/6) - 1}, {vM, vH, 
        t0}]

Solve[{vM t0 == 12, vH (t0 - 5/6) == 12,  
          vM (t0 + 5/6) == vH (t0 + 5/6 - 5/6) - 1}, {vM, vH, t0}] // 
      N

Solve[{(18/5) t1 == 6, vH1 (t1 - 5/6) == 6}, { vH1, 
        t1}]

Solve[{(18/5) t1 == 6, vH1 (t1 - 5/6) == 6}, { vH1, 
          t1}] // N

4

a, b

Remove["Global`*"]

f[a_, x_] := a x^2;  
    t[a_, x_, x0_] := 
      a 2 x0 x + b;    n[a_, x_, x0_] := -1/(a 2 x0) x + 
        c;
bSolv = Flatten[Solve[f[a, x0] == t[a, x0, x0], {b}]]

Position[{x -> y}, y]

Extract[bSolv, Position[{x -> y}, y]][[1]]

Extract[bSolv, Position[{x -> y}, y]][[1]] /. {a -> 
          4}

Extract[bSolv, Position[{x -> y}, y]][[1]] /. {x0 -> 
          4}

t[a_, x_, x0_] := a 2 x0 x - a  x0^2

cSolv = Flatten[Solve[f[a, x0] == n[a, x0, x0], {c}]]

n[a_, x_, 
        x0_] := -1/(a 2 x0) x + ((1 + 2 (a^2) (x0^2))/(2 a))

Plot[{f[0.5, x],  t[0.5, x, 1.5],  n[0.5, x, 1.5]}, {x, 0, 2}, 
        AspectRatio -> Automatic];

A2 = Integrate[f[a, x], {x, 0, x0}]

R = x0 f[a, x0]

A1 = R - A2

A1/A2

A3 = x0 (f[a, x0] +  t[a, 0, x0])/2 - A1

A4 = 
      A1 + x0 (-f[a, x0] +  n[a, 0, x0])/2  //  
        Simplify

A4 // Together

A4/A3 //  Simplify

A4/A3 // Together

c

p1 =  {0, t[a, 0, x0]}

nSolv = Flatten[Solve[ n[a, x, x0] == 0, {x}]]

p4 = {Extract[nSolv, Position[{x -> y}, y]][[1]], 
        0}

tSolv = Flatten[Solve[ t[a, x, x0] == 0, {x}]]

p2 = {Extract[tSolv, Position[{x -> y}, y]][[1]], 
        0}

p7 =  {0, n[a, 0, x0]}

sp[a_, x0_] := Evaluate[(p4 - p1) . (p7 - p2)] // Simplify;  
    sp[a, x0]

Solve[((1 - a x0 + 6 (a^2) (x0^2) - 2 (a^3) (x0^3) + 8 (a^4) (x0^4))/(4 (a^2))) == 0, {x0}]

5

a

Remove["Global`*"]

f[a_, x_, b_] := -a x^2 + b;
bSolv = 
      Flatten[Solve[{f[a, 0, b] == 10, f[a, 10, b] == 0}, {a, 
            b}]]

f[x_] := -1/10  x^2 + 10;

b

A[x1_] := 2 (x1 f[x1]  + (10 - x1) f[x1]/2) //  Simplify;  
    A[x1]

Plot[A[x1], {x1, 0, 10}];

D[A[x1], x1] //  Simplify

Solve[Evaluate[D[A[x1], x1] == 0], {x1}]

N[%]

(Integrate[f[x], {x, -10, 10}] - A[x1]) h

(Integrate[f[x], {x, -10, 10}] - A[x1]) h /. {x1 -> 
          (10/3)}

N[%]

6

Plot[Sin[x], {x, 0, 2  Pi}, 
        AspectRatio -> Automatic];

m = 1/(Pi/2)

x0 = ArcCos[m]

x0 // N

h = Sin[ArcCos[(2/π)]] - m x0

% // N

A = h 2  Pi

% // N

Solve[Sqrt[(Pi/2)^2 + 1^2]/1 == h/x, {x}]

N[%]

laenge1 = Sqrt[(Pi/2)^2 + 1^1] 4

laenge = 
      laenge1 + ((2 (Sqrt[-4 + (π^2)]) - 4 ArcCos[(2/π)])/(π (Sqrt[4 + (π^2)])))

laenge // N

Solve[Sqrt[(Pi/2)^2 + 1^2]/(Pi/2) == h/b, {b}]

N[%]

7

p5[a_,  x0_] := {x0, a x0^2};

p3[a_,  x0_] := {x0, 0};

t[a_, x0_, x_] := a 2 x0 x + b;
Solve[t[a, x0, x0] == a x0^2, {b}]

p1[a_,  x0_] := {0, -a x0^2}

n[a_, x0_, x_] := -1/(a 2 x0) x + 
          b;
Solve[n[a, x0, x0] == a x0^2, {b}]

p7[a_,  x0_] := {0, ((1 + 2 (a^2) (x0^2))/(2 a))}

n[a_, x0_, 
        x_] := -1/(a 2 x0) x + ((1 + 2 (a^2) (x0^2))/(2 a))

Solve[n[a, x0, x] == 0, {x}]

p4[a_,  x0_] := {x0 (1 + 2 (a^2) (x0^2)), 0}

(p7[a,  x0] - p3[a,  x0]) . (p4[a,  x0] - 
          p1[a,  x0])

Solve[-(1/2) (x0^2) (1 + 2 (a^2) (x0^2)) == 
        0, {x0}]